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Controlling Starting Values

To fit the preceding model you must specify a better starting value for C. Avoid starting values of C that are either very large or close to 0. For starting values of A and B, you can specify values, use the default, or have PROC MODEL fit starting values for them conditional on the starting value for C.

Starting values are specified with the START= option of the FIT statement or in a PARMS statement. In PROC MODEL, you have several options to specify starting values for the parameters to be estimated. When more than one option is specified, the options are implemented in the following order of precedence (from highest to lowest): the START= option, the PARMS statement initialization value, the ESTDATA= option, and the PARMSDATA= option. When no starting values for the parameter estimates are specified with BY group processing, the default start value 0.0001 is used for each by group. Again, when no starting values are specified, and a model with a FIT statement is stored by the OUTMODEL=outmodel-filename option in a previous step, the outmodel-filename can be invoked in a subsequent PROC MODEL step by using the MODEL=outmodel-filename option with multiple estimation methods in the second step. In such a case, the parameter estimates from the outmodel-filename are used directly as starting values for OLS, and OLS results from the second step provide starting values for the subsequent estimation method such as 2SLS or SUR, provided that NOOLS is not specified.

For example, the following statements estimate the model parameters by using the starting values A=0.0001, B=0.0001, and C=5.

proc model data=test;
   y = a + b * x ** c;
   label a = "Intercept"
         b = "Coefficient of Transformed X"
         c = "Power Transformation Parameter";
   fit y start=(c=5);
run;

Using these starting values, the estimates converge in 16 iterations. The results are shown in Figure 18.29. Note that since the START= option explicitly declares parameters, the parameter C is placed first in the table.

Using the STARTITER Option

PROC MODEL can compute starting values for some parameters conditional on starting values you specify for the other parameters. You supply starting values for some parameters and specify the STARTITER option on the FIT statement.

For example, the following statements set C to 1 and compute starting values for A and B by estimating these parameters conditional on the fixed value of C. With C=1, this is equivalent to computing A and B by linear regression on X. A PARMS statement is used to declare the parameters in alphabetical order. The ITPRINT option is used to print the parameter values at each iteration.

proc model data=test;
   parms a b c;
   y = a + b * x ** c;
   label a = "Intercept"
         b = "Coefficient of Transformed X"
         c = "Power Transformation Parameter";
   fit y start=(c=1) / startiter itprint;
run;

With better starting values, the estimates converge in only eight iterations. Counting the iteration required to compute the starting values for A and B, this is seven fewer than the 16 iterations required without the STARTITER option. The iteration history listing is shown in Figure 18.30.

Finding Starting Values by Grid Search

PROC MODEL can try various combinations of parameter values and use the combination that produces the smallest objective function value as starting values. (For OLS the objective function is the residual mean square.) This is known as a preliminary grid search. You can combine the STARTITER option with a grid search.

For example, the following statements try five different starting values for C: 1, 0.7, 0.5, 0.3, and 0. For each value of C, values for A and B are estimated. The combination of A, B, and C values that produce the smallest residual mean square is then used to start the iterative process.

proc model data=test;
   parms a b c;
   y = a + b * x ** c;
   label a = "Intercept"
         b = "Coefficient of Transformed X"
         c = "Power Transformation Parameter";
   fit y start=(c=1 .7 .5 .3 0) / startiter itprint;
run;

The iteration history listing is shown in Figure 18.31. Using the best starting values found by the grid search, the OLS estimation only requires two iterations. However, since the grid search required nine iterations, the total iterations in this case is 11.

Guessing Starting Values from the Logic of the Model

Example 18.1, which uses a logistic growth curve model of the U.S. population, illustrates the need for reasonable starting values. This model can be written

where t is time in years. The model is estimated by using decennial census data of the U.S. population in millions. If this simple but highly nonlinear model is estimated by using the default starting values, the estimation fails to converge.

To find reasonable starting values, first consider the meaning of a and c. Taking the limit as time increases, a is the limiting or maximum possible population. So, as a starting value for a, several times the most recent population known can be used—for example, one billion (1000 million).

Dividing the time derivative by the function to find the growth rate and taking the limit as t moves into the past, you can determine that c is the initial growth rate. You can examine the data and compute an estimate of the growth rate for the first few decades, or you can pick a number that sounds like a plausible population growth rate figure, such as 2%.

To find a starting value for b, let t equal the base year used, 1790, which causes c to drop out of the formula for that year, and then solve for the value of b that is consistent with the known population in 1790 and with the starting value of a. This yields or about 5.5, where a is 1000 and 3.9 is roughly the population for 1790 given in the data. The estimates converge using these starting values.

Convergence Problems

When estimating nonlinear models, you might encounter some of the following convergence problems.

Discontinuities

The computational methods assume that the model is a continuous and smooth function of the parameters. If this is not the case, the methods might not work.

If the model equations or their derivatives contain discontinuities, the estimation usually succeeds, provided that the final parameter estimates lie in a continuous interval and that the iterations do not produce parameter values at points of discontinuity or parameter values that try to cross asymptotes.

One common case of discontinuities causing estimation failure is that of an asymptotic discontinuity between the final estimates and the initial values. For example, consider the following model, which is basically linear but is written with one parameter in reciprocal form:

   y = a + b * x1 + x2 / c;

By placing the parameter C in the denominator, a singularity is introduced into the parameter space at C=0. This is not necessarily a problem, but if the correct estimate of C is negative while the starting value is positive (or vice versa), the asymptotic discontinuity at 0 will lie between the estimate and the starting value. This means that the iterations have to pass through the singularity to get to the correct estimates. The situation is shown in Figure 18.32.

Figure 18.32 Asymptotic Discontinuity

Because of the incorrect sign of the starting value, the C estimate goes off towards positive infinity in a vain effort to get past the asymptote and onto the correct arm of the hyperbola. As the computer is required to work with ever closer approximations to infinity, the numerical calculations break down and an "objective function was not improved" convergence failure message is printed. At this point, the iterations terminate with an extremely large positive value for C. When the sign of the starting value for C is changed, the estimates converge quickly to the correct values.

Linear Dependencies

In some cases, the Jacobian matrix might not be of full rank; parameters might not be fully identified for the current parameter values with the current data. When linear dependencies occur among the derivatives of the model, some parameters appear with a standard error of 0 and with the word BIASED printed in place of the t statistic. When this happens, collinearity diagnostics for the Jacobian crossproducts matrix are printed if the DETAILS option is specified and there are twenty or fewer parameters estimated. Collinearity diagnostics are also printed out automatically when a minimization method fails, or when the COLLIN option is specified.

For each parameter, the proportion of the variance of the estimate accounted for by each principal component is printed. The principal components are constructed from the eigenvalues and eigenvectors of the correlation matrix (scaled covariance matrix). When collinearity exists, a principal component is associated with proportion of the variance of more than one parameter. The numbers reported are proportions so they remain between 0 and 1. If two or more parameters have large proportion values associated with the same principal component, then two problems can occur: the computation of the parameter estimates are slow or nonconvergent; and the parameter estimates have inflated variances (Belsley, Kuh, and Welsch 1980, p. 105–117).

For example, the following cubic model is fit to a quadratic data set:

proc model data=test3;
   exogenous x1;
   parms b1 a1 c1 ;
   y1 = a1 * x1 + b1 * x1 * x1  + c1 * x1 * x1 *x1;
   fit y1 / collin;
run;

The collinearity diagnostics are shown in Figure 18.33.

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